A new component is needed for sampling-based planning under differential constraints because of (14.1). Motions are now expressed in terms of an action trajectory, but collision detection and constraint satisfaction tests must be performed in . Therefore, the system, needs to be integrated frequently during the planning process. Similar to the modeling of collision detection as a ``black box,'' the integration process is modeled as a module called the system simulator. See Figure 14.9. Since the systems considered in this chapter are time-invariant, the starting time for any required integration can always be shifted to start at . Integration can be considered as a module that implements (14.1) by computing the state trajectory resulting from a given initial state , an action trajectory , and time . The incremental simulator encapsulates the details of integrating the state transition equation so that they do not need to be addressed in the design of planners. However, that information from the particular state transition equation may still be important in the design of the planning algorithm.